Exploring chaos and bifurcation in a discrete prey–predator based on coupled logistic map

This research paper investigates discrete predator-prey dynamics with two logistic maps. The study extensively examines various aspects of the system’s behavior. Firstly, it thoroughly investigates the existence and stability of fixed points within the system. We explores the emergence of transcritical bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations that arise from coexisting positive fixed points. By employing central bifurcation theory and bifurcation theory techniques. Chaotic behavior is analyzed using Marotto’s approach. The OGY feedback control method is implemented to control chaos. Theoretical findings are validated through numerical simulations.

where x n and y n stand for the densities of prey and predator populations, respectively.The term ax n (1 − x n ) represents the rate of increase of the prey populations in the absence of predator.The term cx n y n stands for the rate of decrease due to predation, where the parameter c is the predation parameter.The term by n (1 − y n ) represents the rate of decrease in predator populations in the absence of prey.The term dx n y n denotes the growth rate of the predator in the presence of the prey.The coefficients a, b, c and d are positive constants, and 0 < x n , y n < 1, 0 < a ≤ 4 , and 0 < b ≤ 4 are the coupling parameters.The term a x n represents the exponential growth of the prey.The term a x 2 n represents the struggle of preys for food (mating competition between prey males or control of herd leadership).The term c x n y n stands for the rate of decrease due to predation.The term b y n represents the exponential growth of predator.The term b y 2 n represents the struggle of predators over prey (mating competition between predatory males or control of herd leadership).
It is crucial to highlight that although numerical analysis has been conducted on this system 23,24 , a significant portion of its potential dynamic behaviors, including transcritical bifurcation, period-doubling bifurcation, Neimark-Sacker bifurcation, and others, remain unexplored analytically.In this study, we employ coupled logistic map modeling and difference equations to qualitatively analyze a discrete-time prey-predator model.Our primary objective is to investigate the model's behavior by exploring diverse parameter values and initial conditions.Specifically, we focus on the emergence of stable equilibria, period doubling, chaotic attractors, as well as the analysis of codimension one bifurcations and chaos control.The results of our analysis reveal a wide range of captivating dynamic behaviors, showcasing the model's heightened sensitivity to variations in key parameters. (1) These findings hold significant implications for our comprehension of prey-predator systems, underscoring the importance of qualitative analysis in unraveling the complexities inherent in ecological systems.We demonstrate that the model (1) undergoes various types of codimension-one bifurcations through analytical methods.We present bifurcation diagrams and phase portraits to illustrate these bifurcations.Our analysis reveals a multitude of complex and diverse dynamic behaviors, including the presence of limit cycles, periodic solutions, and chaos.These findings highlight the intricate and rich nature of the system's dynamics.
This paper is structured as follows: "Analysis of fixed points and their stability" focuses on discussing the stability of the model's fixed points.In "Local bifurcations analysis" , we delve into the transcritical bifurcation, period-doubling, and Neimark-Sacker bifurcation, providing a comprehensive analysis of bifurcations at the positive fixed point."Existence of Marotto's chaos" explores the conditions indicating the presence of Marotto's chaos.The utilization of chaos control strategy to manage the chaotic behavior of the model (1) is presented in "Chaos control" .For clarity, numerical simulations are provided in "Numerical simulations" to illustrate the key findings.Finally, in "Conclusion" , we summarize our primary discoveries and draw conclusions as we conclude the paper.

Analysis of fixed points and their stability
In this section, we provide qualitative properties for all fixed points in model (1) as well as the conditions for fixed point asymptotic stability.We have four fixed points, as follows: (i) When the initial conditions are p 0 (0, 0) , the total population undergoes extinction.(ii) In this scenario, the prey becomes extinct, or in the case of the predator only, when p 1 a−1 a , 0 , where a > 1.
(iii) When the initial conditions are p 2 (0, b−1 b ) , with b > 1 , the predator becomes extinct, or in the case of the prey only.(iv) In the case of cohabitation, both the prey and predator coexist when the initial conditions are The model (1) is rewritten as follows: The Jacobian matrix (J) associated with model (2) at point p(x, y) can be expressed as follows: where The auxiliary polynomial is formulated as follows: where the quadratic equation ( 4) has one variable T(x, y) = −(j 11 + j 22 ) and D(x, y) = j 11 j 22 − j 12 j 21 .
Lemma 1 25,26 .Consider the function F (R) = R 2 + TR + D , where R 1 and R 2 are the two roots of the equation F (R) = 0 .Assuming that F (1) > 0 , we have the following: The conditions R 1 and R 2 being complex, and Definition 1 25,26 .The fixed point p(x, y) is called (3) J(x, y) = j 11 j 12 j 21 j 22 ,

Proof
The Jacobian matrix J at p 0 (0, 0) has the following form: which have two eigenvalues: R 1 = a and R 2 = b .Obviously, by applying Lemma 1 and Definition 1. Look at Figs. 1(i) and 3.

Theorem 2
The aixel fixed point p 1 a−1 a , 0 exhibits the following characteristics: Proof The Jacobian matrix J at p 1 ( a − 1 a , 0) has the following form: (5) Proof The Jacobian matrix J evaluated at , then P 3 (x, y) has the following topological properties 25,26 : Bifurcation analysis of the fixed point P 0 (0, 0) The Jacobian matrix evaluated at P 0 (0, 0) is given by: If a = 1 , then J(P 0 ) has two eigenvalues: R 1 = 1 and R 2 = b .If b = 1 , then |R 2 | � = 1 so the conditions of the appearance of transcritical bifurcation at P 0 is represented by the following Theorem: is subject to a transcritical bifurcation at P 0 (0, 0) and has only one fixed point.
Proof By introducing the new dependent variable σ n = a − 1 , the model (1) can be transformed into: where Let We restricted the system (6) to center manifold s(x n , σ n ) under the condition Thus, we can get the coefficients of the center manifold as follows: Thus, the center manifold takes the following form: The map is restricted to the central manifold, which is defined by the following expression: Here, we define non-zero real numbers as follows: The model (1) undergoes a transcritical bifurcation at P 0 (0, 0) .This completes the proof.
Theorem 6 If a = −1 , b = 1 , then the model (1) is subject to a period-doubling bifurcation at P 0 (0, 0) and has only one fixed point.
Proof If we introduce the new dependent variable σ n = a + 1 , the model (1) can be expressed as follows: where Let We restricted the system (11) to center manifold s(x n , σ n ) under the condition Thus, we can get the coefficients of the center manifold as follows: Therefore, the center manifold can be characterized by the following expression: The dynamics of the map are confined to the central manifold, which is described by the following expression: Since and ( 7) (8) δ1 = 0, δ2 = 0 and δ3 = 0.

Bifurcation analysis of the fixed point P 1
a−1 a , 0 The Jacobian matrix at 1 so the conditions of the appearance of transcritical bifurcation at P 1 is summarized by the following theorem: and has only one fixed point.

Proof
Vol.:(0123456789)And the map is limited to the central manifold that was given by Since The model (1) undergoes a transcritical bifurcation at P 1 ( a − 1 a , 0) .This completes the proof.
is subject to a period-doubling bifurcation at ) and has only one fixed point.The model (1) undergoes a period-doubling bifurcation at P 1 ( a − 1 a , 0) .This completes the proof.

Bifurcation analysis of the fixed point P 2 0, b−1 b
The Jacobian matrix at 1 so the conditions of the appearance of transcritical bifurcation at P 2 is represented by the following theorem.( ) .This completes the proof.
is subject to a period-doubling bifurcation at  ) .This completes the proof.

Bifurcation analysis of P 3 ( x, y)
In this subsection, we investigate the bifurcation analysis centered on P 3 (x, y) .We consider that and To explore the period-doubling bifurcation of model (1) at P 3 (x, y) when its parameters change in the neighbor- hood of F P 3 , we study the model (1) for (a, b, c, d) ∈ F P 3 as follows: The eigenvalues for P 3 (x, y) of the model (40) are R 1 = −1 and |R 2 | � = 1 .By selecting a * as the bifurcation parameter and introducing a small perturbation a 1 to the Eq. (40),we obtain: where a * ≪ 1.
Let u = x − x , u = y − y .Then system (41) converted to where (37) (40) Applying the center manifold theorem 27 , we obtain: Using the same technique used with the other fixed points, the center manifold map is generated as follows: where Let and (43) (44) Theorem 11 If 1 = 0 and 2 = 0 , then model (1) undergoes a period-doubling bifurcation at the unique positive fixed point P 3 (x, y) when the parameter a varies within a small neighborhood of F P 3 .Additionally, when � 2 > 0 (respectively, � 2 < 0 ), the period-2 orbits that arise from P 3 (x, y) exhibit stability (respectively, instability).Now, let's examine the Neimark-Sacker bifurcation of P 3 (x, y) when the parameters (a, b, c, d) vary within a small neighborhood of N P 3 .By selecting parameters (a, b, c, d) ∈ N P 3 , we can represent the model (1) as follows: We have parameter (a, b, c, d) . Choosing a 2 as a bifurcation parameter and little disturb the model (47) by ā * ≪ 1 to get Let u = x − x , u = y − y .Then model (47) converted to the values of Ê11 , Ê12 , Ê13 , Ê14 , Ê21 , Ê22 , Ê23 , Ê14 are obtained from Eq. ( 43) by substituting a with a 2 + ā * .The characteristic equation of the Jacobian matrix evaluated at (u, v) = (0, 0) can be written as: where and As the parameters (a, b, c, d) belong to the neighborhood N P 3 , the eigenvalues of p 3 (x, y) can be expressed as a pair of complex conjugate numbers R and R , with a modulus given by Eq. (1) according to Theorem 4. Here, we have:

Existence of Marotto's chaos
In this section, we establish that the model (1) displays chaotic behavior, as defined by Marotto 31,32 .
Definition 2 (42) Let's consider the function F : R µ → R µ , which is differentiable in the neighborhood B r (Z) .
We define Z ∈ R as an expanding fixed point of F in B r (Z) if F (Z) = Z and all eigenvalues of the Jacobian matrix DF(X ) have a magnitude greater than 1 for all X ∈ B r (Z).
Definition 3 (42) Suppose Z is an expanding fixed point of F in B r (Z) for a positive value r.In such a case, Z is considered a snapback repeller of F if there exists a point X 0 ∈ B r (Z) with X 0 = Z , F η (X 0 ) = Z , and DF η (X 0 ) = 0 for a positive integer η.
Next, an example is provided to demonstrate how to fulfil the conditions outlined in Theorem 13.

Chaos control
The utilisation of different bifurcation parameters enhances our ability to comprehensively analyse the system's behavior.By selecting independent parameters, we can isolate their influences and gain insights into their effects on the system's dynamics.This approach allows us to examine the system's sensitivity to various parameters, identify those that have a significant impact on bifurcations, and develop a deeper understanding of its behavior under diverse conditions.Choosing multiple parameters makes the study practical and feasible, as some of them may be controlled or measured in experiments.Furthermore, effective ecological conservation management solutions can be devised to preserve these intricate ecosystems.Chaotic dynamics within a system can lead to instabilities and undesirable behaviors.Therefore, effective techniques for chaos control are crucial for mitigating harmful chaotic behaviors.To improve system performance, it is necessary to regulate chaotic dynamics towards a periodic orbit or a fixed point.Chaos control involves managing chaotic dynamics within complex nonlinear systems, and the literature offers numerous strategies for achieving this, such as the Ott-Grebogi-Yorke (OGY) method 10,33,34 and feedback control methods applied to model (1).For controlling chaos arising from Neimark-Sacker and period-doubling bifurcations at the fixed point of model (57), we employ the OGY technique, resulting in the following formulation of model (57): We introduce the parameter c as a chaos control parameter, with c belonging to the interval (c 0 − δ, c 0 + δ) , where δ > 0 and c 0 represents the nominal value of c.Additionally, we consider p 3 (x, y) as the fixed point of model (1).In the vicinity of the fixed point p 3 (x, y) , model (57) can be approximated as follows: where (55) x 2 = 1 dy 2 (by 2 2 − by 2 + y), (56) x 1 = 1 dy 1 (by 1 2 − by 1 + y 2 ), (57) and Furthermore, the controlled model (57) can be represented by the following matrix if the condition holds: , where = 1 2 , then system (58) can be written as Furthermore, the fixed point p 3 (x, y) is considered locally stable if and only if both eigenvalues of the matrix J − L lie within an open unit disk.The matrix J − L can be written as follows: where The auxiliary equation of the Jacobian matrix J − L is: Let R 1 and R 2 are the eigenvalues of characteristic equation ( 59), then we have and Furthermore, we assume R 1 = ±1 and R 1 R 2 = 1 .Consequently, the lines of marginal stability for (60) and (61) can be calculated as follows: Next, we suppose that R 1 = 1 , then (61) and (60) yield that Finally, if R 1 = −1 and we use (60) and (61), we get Hence, the stability region of (57) forms a triangular region bounded by L 1 , L 2 , and L 3 in the 1 2 plane.

Numerical simulations
In this section, we utilize numerical simulations to validate our theoretical findings and illustrate the complex dynamical behaviors of system (1).We present bifurcation diagrams, phase portraits, and compute the maximum Lyapunov exponents (referred to as ML) to provide a comprehensive understanding of the system's behavior. , (62) (64) L 3 : j 21 2 − 1 (j 22 + 1) � + j 11 + 1 j 22 − j 12 j 21 + j 11 + 1 = 0.In Fig. 3, we illustrate the relationship between generation n and the populations x n of the prey and y n of the predator to assess their qualitative behavior.In Fig. 3(i), all population curves for the prey x n and the predator y n tend towards zero as n increases, indicating the extinction of both prey and predator when the conditions 0 < a = 0.5 < 1 and 0 < b = 0.8 < 1 are met.This implies that p 0 (0.00001, 0.00001) is stable.In Fig. 3(ii), where a = 1.3 > 1, b = 0.8 < 1 , the prey's x n curve goes from zero to infinity, while the prey's y n curve equals zero.
Similarly, in Fig. 3(iii), where a = 0.9 < 1 , b = 1.8 > 1 , the prey curve x n equals zero, while the predator's y n curve goes from zero to infinity, which is unacceptable.Finally, in Fig. 3(iv), where a = 1.9 > 1 , b = 1.5 > 1 , it's evident that as the x n curve for prey decreases, the y n curve for the predator increases.This presents a contradic- tion because the parameters a and b do not satisfy the stability conditions.
(II) Examine parameters a and b while keeping the remaining parameters constant at c = 0.6 and d = 0.5 .In Fig. 4, we illustrate the relationship between generation n and the populations x n of prey and y n of predators to evaluate their specific behavior at P 1 ( a − 1 a , 0) .In Fig. 4(i), we observe that all population curves for prey x n and predator y n converge to 0.5 and 0, respectively, as n progresses.This indicates the extinction of the predator and the stability of the prey when the conditions 1 < a = 2 < 3 and 0 < b = 0.7 < a+d−ad a are satisfied.This suggests that the P 1 (0.5, 0.00001) is in a stable state.In Fig. 4(ii), where a = 3.2 > 3, b = 0.5 < a+d−ad a , we observe that both the prey and the predator at (0.6875, 0.00001) with n = 30 are in a stable state.As n increases, we observe that the prey has moved to a periodic state.In this scenario, it becomes unstable, indicating regular prey growth with the predator remaining consistently at zero.In Fig. 4(iii), with a = 2.5 < 3 and b = 0.9 > a+d−ad a , it becomes apparent that the xn curve for prey is decreasing, while the yn curve for the predator is increasing.This contradiction arises because the parameters a and b fail to meet the stability conditions.We find that the description of the situation in Fig. 4(iv), where a = 3.8 > 3, b = 0.68 > a+d−ad a , at point (0.7368, 0.00001) is similar to the description in Fig. 3(ii), except that the growth of the prey reaches a stage of chaos and uncontrollability in the absence of the predator.(III) As in the previous case (II), delete the explanation.(IV) Analyse the parameters a and b, keeping the other parameters constant at c = 0.7 and d = 0.73.In Fig. 5, we illustrate the relationship between generation n and population x n of prey and y n of predators to evaluate their distinct behavior at p 3 (x, y) .In Fig. 5(i), we observe that, over time, all population curves for prey x n and predator y n converge to 0.346170489 and 0.252704457, respectively.This indicates that both the predator and prey are stable when the conditions a = 1.8 and b = 1 are satisfied, with the initial values (0.2, 0.1) in a steady state.In Fig. 5(ii), where a = 3.6 and b = 1.4,we observe that both the prey and the predator at p 3 (0.6875, 0.00001) with the initial values (0.5, 0.79) are in an unstable state.We notice that both predator and prey have moved to a cyclic state, which indicates regular growth for both predator and prey.We also employ different values for the parameters a = 3.7, 3.8 and b = 2.3, 2.8 in Fig. 5(iii), (iv), respectively, resulting in curves that do not converge to the initial values (0.5, 0.79).We note two observations.First, when the parameter a (which indicates the growth of the prey) increases by a greater value than the parameter b (which indicates the growth of the predator), this leads to an increase in the number of predators.Second, an increase in the number of prey is matched by an increase in predation, which leads to an increase in the number of prey.

Conclusion
This model represents predator-prey interactions that play a vital role in maintaining stability and biodiversity in an ecosystem.An increased population density of prey leads to instability in the ecosystem in the absence of predators.The opposite is true: in the absence of prey, this leads to the extinction of predators and, thus, the collapse of the ecosystem.Predation is a directly proportional relationship between predator and prey; when prey is abundant, predators increase, and thus the number of prey through predation decreases.Conversely, when predators are abundant, prey is scarce, and thus the number of predators decreases through starvation, intensifying the struggle for prey.(Mating between males is usually an aggressive act towards females that sometimes results in serious injury or death, but in those species in which several females become fertile at the same time, there can be a strong element of competition and dominance for access to females for reproduction).In this study, we employed coupled logistic maps and difference equations to qualitatively analyse a discrete-time prey-predator model.Our main objective was to explore the model's behaviour by investigating various parameter values and initial conditions.We focused on the emergence of stable equilibria, period doubling, and chaotic attractors, as well as the analysis of codimension-one bifurcations, including Marotto's chaos and chaos control.Our analysis revealed a wide range of captivating dynamic behaviours, highlighting the model's sensitivity to variations in key parameters.These findings have significant implications for our understanding of prey-predator systems.The bifurcation phenomena in such systems can be studied using the center manifold theorem.In this paper, we took a step towards understanding the dynamics of these systems by investigating local and global bifurcations in a two-dimensional normal form map derived from the proposed model.We conducted a detailed investigation of the nature of eigenvalues and calculated period-doubling and Neimark-Sacker bifurcations.We also derived conditions for the occurrence of local codimension-one bifurcations.To demonstrate the occurrence of various bifurcations in this map, we presented numerical examples.Overall, our study contributes to the understanding of the dynamic behaviours exhibited by prey-predator models and provides insights into the occurrence of different bifurcations in such systems.By uncovering the intricate dynamics of these systems, the research contributes to our broader knowledge of ecological systems and their behaviour.
In the future, a deeper understanding of ecosystem stability can be attained through the exploration of the global dynamics of the predator-prey model.By systematically varying parameters and employing codimension-2 bifurcation analysis, we can uncover novel bifurcation phenomena and their implications for population dynamics.Additionally, in light of the substantial influence of the Allee effect on predator populations, future research endeavors can prioritize its investigation.

Theorem 4
p 2 (0, b−1 b ) takes the following form: which has two eigenvalues: R 1 = 2 − b and R 2 = ab − bc + c b .Clearly, by employing Lemma 1 and Definition 1, the results can be observed in Fig. 1(iii) if c = 0.25.The characteristic equation at P 3 (x, y) has the form: If b(c + 1) < ab + c or d + a < a(b + d)
By introducing the new dependent variables σ n = b − a+d−ad a , α n = x n − a−1 a , and β n = y n , the model (1) can be transformed into: 1 a and β n = y n be a new dependent variable.Then,
can get that And the map is limited to the central manifold that was given by Since The model (1) undergoes a transcritical bifurcation at P 2 (0, b − 1 b

P 2
(0, b − 1 b ) and has only one fixed point.Proof By introducing the new dependent variables σ n = a − bc−b−c b , α n = x n , and β n = y n − b−1 b , the model (1) can be transformed into: Let and use the transformation system (34) becomes where (31)

Figure 3 .
Figure 3. Behavioral characteristics of prey populations x n and predator populations y n .

Figure 7 (
iii) illustrates the corresponding maximum Lyapunov exponents from Fig.7(i), (ii).(III)When taking the parameters b = 3.5, a = 2 , and d = 0.7 , we observe from Fig.8(i), (ii) that the fixed point p 3 (x, y) remains stable within the range 1.455 < c ≤ 1.6 and loses its stability at the value c = 1.455,

Figure 4 .
Figure 4. Behavioral characteristics of prey populations x n and predator populations y n .

Figure 5 .
Figure 5. Behavioral characteristics of prey populations x n and predator populations y n .